Question

1. a circle has a radius of 8 inches which is changing. write the following formulas.

circumference
$c = 2\pi r$
or
$c = 16\pi$
change of circumference
area
change of area

Explanation:

Step1: Recall the formula for the area of a circle

The formula for the area \(A\) of a circle is \(A=\pi r^{2}\), where \(r\) is the radius of the circle.

Step2: Differentiate the area formula with respect to time \(t\)

Using the chain - rule \(\frac{dA}{dt}=\frac{d}{dt}(\pi r^{2})\). Since \(\pi\) is a constant, \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)

Step3: Recall the formula for the circumference of a circle (already partially done)

The formula for the circumference \(C\) of a circle is \(C = 2\pi r\).

Step4: Differentiate the circumference formula with respect to time \(t\)

Differentiating \(C = 2\pi r\) with respect to \(t\) (using the chain - rule, since \(r\) is a function of \(t\)). Since \(2\pi\) is a constant, \(\frac{dC}{dt}=2\pi\frac{dr}{dt}\)

Answer:

Circumference: \(C = 2\pi r\)
Change of Circumference: \(\frac{dC}{dt}=2\pi\frac{dr}{dt}\)
Area: \(A=\pi r^{2}\)
Change of Area: \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)